This study uses HI image data from the Widefield ASKAP L-band Legacy All-sky Blind surveY (WALLABY) pilot survey with the Australian Square Kilometre Array Pathfinder (ASKAP) telescope, covering the Hydra cluster out to 2.5r_200_. We present the projected phase-space distribution of HI-detected galaxies in Hydra, and identify that nearly two-thirds of the galaxies within 1.25r_200_ may be in the early stages of ram pressure stripping. More than half of these may be only weakly stripped, with the ratio of strippable HI (i.e., where the galactic restoring force is lower than the ram pressure in the disk) mass fraction (over total HI mass) distributed uniformly below 90%. Consequently, the HI mass is expected to decrease by only a few 0.1dex after the currently strippable portion of HI in these systems has been stripped. A more detailed look at the subset of galaxies that are spatially resolved by WALLABY observations shows that, while it typically takes less than 200Myr for ram pressure stripping to remove the currently strippable portion of HI, it may take more than 600Myr to significantly change the total HI mass. Our results provide new clues to understanding the different rates of HI depletion and star formation quenching in cluster galaxies.